A Unified Algorithmic Framework for Nested q-Binomial and Binomial Sums

We introduce a unified framework for computing nested sums involving products of q-binomial coefficients and q-integers, and organize their evaluation into explicit calculation formulas. The core of the framework is that every such sum admits three equivalent expansions in q-binomial coefficients, linked by explicit transformations. These transformations yield inversion formulas, orthogonality relations, and closed-form evaluations for weighted infinite series. The framework recovers many classical q-binomial theorems as special cases and systematically generates new summation formulas through parameter specialization. In the classical limit q → 1, the machinery reduces to ordinary binomial sums. It gives explicit formulas for the Stirling numbers of both kinds (with parameters N and N − M) and for the Eulerian numbers. In the modular setting, the same formalism provides a unified treatment of a broad family of congruences for nested sums modulo primes.